What is the general solution for (r-c)=asqrt(2)sqrt(sin(θ))+d ?

The general solution for (r-c)=asqrt(2)sqrt(sin(θ))+d is

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(r-c)=asqrt(2)sqrt(sin(θ))+d

Solution

(r - c) = a 2 sin (θ) + d

Solution

θ = arcsin (- \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}) + 2 πn, θ = π + arcsin (\frac{rc}{a ^{2}} + \frac{r d}{a ^{2}} - \frac{r ^{2}}{2 a ^{2}} - \frac{c ^{2}}{2 a ^{2}} - \frac{c d}{a ^{2}} - \frac{d ^{2}}{2 a ^{2}}) + 2 πn

Solution steps

(r - c) = a 2 sin (θ) + d

Switch sides

a 2 sin (θ) + d = r - c

Move

d

to the right side

a 2 sin (θ) + d = r - c

Subtract

d

from both sides

a 2 sin (θ) + d - d = r - c - d

Simplify

a 2 sin (θ) = r - c - d

a 2 sin (θ) = r - c - d

Divide both sides by

a 2

a 2 sin (θ) = r - c - d

Divide both sides by

a 2

\frac{a 2 sin ( θ )}{a 2} = \frac{r}{a 2} - \frac{c}{a 2} - \frac{d}{a 2}

Simplify

\frac{a 2 sin ( θ )}{a 2} = \frac{r}{a 2} - \frac{c}{a 2} - \frac{d}{a 2}

Simplify

\frac{a 2 sin ( θ )}{a 2} : sin (θ)

\frac{a 2 sin ( θ )}{a 2}

Cancel the common factor:

a

= \frac{2 sin ( θ )}{2}

Cancel the common factor:

2

= sin (θ)

Simplify

\frac{r}{a 2} - \frac{c}{a 2} - \frac{d}{a 2} : \frac{2 ( r - c - d )}{2 a}

\frac{r}{a 2} - \frac{c}{a 2} - \frac{d}{a 2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{r - c - d}{2 a}

Multiply by the conjugate

\frac{2}{2}

= \frac{( r - c - d ) 2}{a 2 2}

a 22 = 2 a

a 22

Apply radical rule:

a a = a

22 = 2

= 2 a

= \frac{2 ( r - c - d )}{2 a}

sin (θ) = \frac{2 ( r - c - d )}{2 a}

sin (θ) = \frac{2 ( r - c - d )}{2 a}

sin (θ) = \frac{2 ( r - c - d )}{2 a}

Square both sides:

sin (θ) = - \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

sin (θ) = \frac{2 ( r - c - d )}{2 a}

(sin (θ))^{2} = (\frac{2 ( r - c - d )}{2 a})^{2}

Expand

(sin (θ))^{2} : sin (θ)

(sin (θ))^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (sin^{\frac{1}{2}} (θ))^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= sin^{\frac{1}{2} \cdot 2} (θ)

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= sin (θ)

Expand

(\frac{2 ( r - c - d )}{2 a})^{2} : - \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

(\frac{2 ( r - c - d )}{2 a})^{2}

\frac{2 ( r - c - d )}{2 a} = \frac{2 r - 2 c - 2 d}{2 a}

\frac{2 ( r - c - d )}{2 a}

Expand

2 (r - c - d) : 2 r - 2 c - 2 d

2 (r - c - d)

Distribute parentheses

= 2 r + 2 (- c) + 2 (- d)

Apply minus-plus rules

+ (- a) = - a

= 2 r - 2 c - 2 d

= \frac{2 r - 2 c - 2 d}{2 a}

= (\frac{2 r - 2 c - 2 d}{2 a})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 r - 2 c - 2 d ) ^{2}}{( 2 a ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2 a)^{2} = 2^{2} a^{2}

= \frac{( 2 r - 2 c - 2 d ) ^{2}}{2 ^{2} a ^{2}}

(2 r - 2 c - 2 d)^{2} = - 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2}

(2 r - 2 c - 2 d)^{2}

(2 r - 2 c - 2 d)^{2} = (2 r - 2 c - 2 d) (2 r - 2 c - 2 d)

= (2 r - 2 c - 2 d) (2 r - 2 c - 2 d)

Expand

(2 r - 2 c - 2 d) (2 r - 2 c - 2 d) : - 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2}

(2 r - 2 c - 2 d) (2 r - 2 c - 2 d)

Distribute parentheses

= 2 r 2 r + 2 r (- 2 c) + 2 r (- 2 d) + (- 2 c) 2 r + (- 2 c) (- 2 c) + (- 2 c) (- 2 d) + (- 2 d) 2 r + (- 2 d) (- 2 c) + (- 2 d) (- 2 d)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 22 rr - 22 rc - 22 r d - 22 rc + 22 cc + 22 c d - 22 r d + 22 c d + 22 dd

Simplify

22 rr - 22 rc - 22 r d - 22 rc + 22 cc + 22 c d - 22 r d + 22 c d + 22 dd : - 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2}

22 rr - 22 rc - 22 r d - 22 rc + 22 cc + 22 c d - 22 r d + 22 c d + 22 dd

Group like terms

= - 22 rc - 22 rc - 22 r d - 22 r d + 22 rr + 22 cc + 22 c d + 22 c d + 22 dd

Add similar elements:

- 22 rc - 22 rc = - 222 rc

= - 222 rc - 22 r d - 22 r d + 22 rr + 22 cc + 22 c d + 22 c d + 22 dd

Add similar elements:

- 22 r d - 22 r d = - 222 r d

= - 222 rc - 222 r d + 22 rr + 22 cc + 22 c d + 22 c d + 22 dd

Add similar elements:

22 c d + 22 c d = 222 c d

= - 222 rc - 222 r d + 22 rr + 22 cc + 222 c d + 22 dd

222 rc = 4 rc

222 rc

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 2 rc

Multiply the numbers:

2 \cdot 2 = 4

= 4 rc

222 r d = 4 r d

222 r d

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 2 r d

Multiply the numbers:

2 \cdot 2 = 4

= 4 r d

22 rr = 2 r^{2}

22 rr

Apply radical rule:

a a = a

22 = 2

= 2 rr

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

rr = r^{1 + 1}

= 2 r^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 r^{2}

22 cc = 2 c^{2}

22 cc

Apply radical rule:

a a = a

22 = 2

= 2 cc

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cc = c^{1 + 1}

= 2 c^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 c^{2}

222 c d = 4 c d

222 c d

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 2 c d

Multiply the numbers:

2 \cdot 2 = 4

= 4 c d

22 dd = 2 d^{2}

22 dd

Apply radical rule:

a a = a

22 = 2

= 2 dd

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

dd = d^{1 + 1}

= 2 d^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 d^{2}

= - 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2}

= - 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2}

= - 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2}

= \frac{( - 4 rc - 4 r d + 2 r ^{2} + 2 c ^{2} + 4 c d + 2 d ^{2} )}{2 ^{2} a ^{2}}

Remove parentheses:

(- a) = - a

= \frac{- 4 rc - 4 r d + 2 r ^{2} + 2 c ^{2} + 4 c d + 2 d ^{2}}{2 ^{2} a ^{2}}

Factor

- 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2} : 2 (- 2 rc - 2 r d + r^{2} + c^{2} + 2 c d + d^{2})

- 4 rc - 4 r d + 2 r^{2} + 2 c^{2} + 4 c d + 2 d^{2}

Rewrite as

= - 2 \cdot 2 rc - 2 \cdot 2 r d + 2 r^{2} + 2 c^{2} + 2 \cdot 2 c d + 2 d^{2}

Factor out common term

2

= 2 (- 2 rc - 2 r d + r^{2} + c^{2} + 2 c d + d^{2})

= \frac{2 ( - 2 rc - 2 r d + r ^{2} + c ^{2} + 2 c d + d ^{2} )}{2 ^{2} a ^{2}}

Cancel the common factor:

2

= \frac{- 2 rc - 2 r d + r ^{2} + c ^{2} + 2 c d + d ^{2}}{2 a ^{2}}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 2 rc - 2 r d + r ^{2} + c ^{2} + 2 c d + d ^{2}}{2 a ^{2}} = - \frac{2 rc}{2 a ^{2}} - \frac{2 r d}{2 a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{2 c d}{2 a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

= - \frac{2 rc}{2 a ^{2}} - \frac{2 r d}{2 a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{2 c d}{2 a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

sin (θ) = - \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

sin (θ) = - \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

Apply trig inverse properties

sin (θ) = - \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

General solutions for

sin (θ) = - \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

θ = arcsin (- \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}) + 2 πn, θ = π + arcsin (\frac{rc}{a ^{2}} + \frac{r d}{a ^{2}} - \frac{r ^{2}}{2 a ^{2}} - \frac{c ^{2}}{2 a ^{2}} - \frac{c d}{a ^{2}} - \frac{d ^{2}}{2 a ^{2}}) + 2 πn

θ = arcsin (- \frac{rc}{a ^{2}} - \frac{r d}{a ^{2}} + \frac{r ^{2}}{2 a ^{2}} + \frac{c ^{2}}{2 a ^{2}} + \frac{c d}{a ^{2}} + \frac{d ^{2}}{2 a ^{2}}) + 2 πn, θ = π + arcsin (\frac{rc}{a ^{2}} + \frac{r d}{a ^{2}} - \frac{r ^{2}}{2 a ^{2}} - \frac{c ^{2}}{2 a ^{2}} - \frac{c d}{a ^{2}} - \frac{d ^{2}}{2 a ^{2}}) + 2 πn

Popular Examples

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2 cos^{2} (t) = sin^{2} (t)

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tan (x) = - 0.35

Frequently Asked Questions (FAQ)

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